Formulas for a Gaussian kernel and a normal probability distribution

Good morning!

This my first post in this forum.
I would very much appreciate your attention.
My question is regarding 1) the proper Gaussian kernel formula and 2) the proper normal distribution formula that I should use in my probem.

I'm going to describe the context and in the end I formulate my question.

THE CONTEXT
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I have to compute "a posteriori" probabilities Ph|u based on a priori training set of Ns samples (Ns sample objects q and corresponding similar objects v obtained experimentally).
The prior distribution can be modeled by a multivariate Normal distribution with conditional mean u(q) and conditional covariance matrix E(q) Pv|q=N(u(q),E(q))
Where N() is the distribution function.

I also have a function h(v) being a linear application of v, wich also produces a Gaussian random variable with conditional covariance matrix aT*E(q)*a and conditional mean h(u(q)). Where aT is the transpose of a. a is known. The distribution of h is then Ph|q=N(h(u(q)),aT*E(q)*a) .
Where N() is the distribution function. And aT the transpose of a. (a is known).

At this point, the conditional functions u(q) and E(q) for any q could be estimated by a Parzen window over the Ns samples of the training set. However, this process would be too computationally expensive.
The problem is then simplified by considering a simpler model where the mean and variance of Ph|q are conditioned only by h(q) and not q itself. The main fundation of this assumption is to consider h(v) independent from h(q) which is quite realistic due to the independence of the vector a.

For each sample object qs we estimate the sample mean u(qs) and sample covariance matrix E(qs) over the similar objects v (We first find out experimentally these v's. Then we calculated the mean and variance over these v's.). For each sample object qs we then compute h(qs) and associate them a sample variance and a sample variance: variance[h(qs)]=aT*Es*a And a sample mean u[h(qs)]=h[u(qs)]

The conditional mean u[h(q)] and conditional variance E[h(q)] for any q are then interpolated by a Gaussian Kernel over the Ns sample: u[h(q)]=SUM{ K[h(q),h(qs)]*h[u(qs)] } / SUM{ K[h(q),h(qs)]

E[h(q)] = SUM{ K[h(q),h(qs)]*h[Es(qs)] } / SUM{ K[h(q),h(qs)]

When SUM goes from s=0 to s=Ns .
Where K(x,y) is the Gaussian kernel funtion and o = 0.2 is the gaussian kernel standard deviation parameter (sigma).

QUESTION 1
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1) Therefore, what should the Gaussian kernel look like? I believe it's a 2d Gaussian Kernel.
PS: I believe it is f(x,y) = 1*e^((-1)*sigma*((x-0)^2+(y-0)^2)), am I right?

QUESTION 2
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The a posteriori probabilitie of finding objects v similar to a specified object q in a u is then:
P[h(q),u]=INTEGRAL{ N( u[h(q)], E[h(q)] ) } dy
Where the INTEGRAL is defined from y=u to y=u+1 .

My question is regarding the formula to calculate this integral and the formula for this discrete distribution N() .

a) PS: Am I right if I do the following?
P[h(q),u]=f(u+1)-f(u)
b) How should N() look like?
Is it the 1d Gaussian : X ~ N(mean,variance)
or the 2d Gaussian: X,Y ~ N(mean,variance)
How does this N look like?